Finite Group Databasebeta

← #2408×C197 #38739342

Subgroups — #2408×C197 #38739342

Order 25216.

1232 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×D4×C197C211yesmaximal, characteristic, complemented
2C2×#121×C197C212yesmaximal, complemented
3C2×#120×C197C212yesmaximal, complemented
4C2×C4×D4×C197C211yesmaximal, characteristic, complemented
5#394×C197C212yesmaximal, complemented
6#392×C197C212yesmaximal, complemented
7#390×C197C212yesmaximal, complemented
8#380×C197C211yesmaximal, characteristic, complemented
9#379×C197C212yesmaximal, complemented
10C25×C197C2212yes—
11C22×D4×C197—28no—
12C22×D4×C197—24no—
13C22×D4×C197C2212yes—
14C22×D4×C197C2212yes—
15C22×D4×C197C2211yescharacteristic
16C22×D4×C197C2214yescomplemented
17C23×C4×C197C2211yescharacteristic
18C23×C4×C197C2211yescharacteristic, complemented
19#121×C197—28no—
20#120×C197—28no—
21C4×D4×C197—24no—
22C2×C4⋊C4×C197C2212yescomplemented
23C2×C4⋊C4×C197C2211yescharacteristic, complemented
24C2×C22⋊C4×C197C2211yescharacteristic
25C2×C22⋊C4×C197C2212yes—
26C2×C22⋊C4×C197C2214yescomplemented
27C2×C22⋊C4×C197C2214yescomplemented
28C2×C22⋊C4×C197C2211yescharacteristic, complemented
29C2×C22⋊C4×C197C2214yescomplemented
30C2×C42×C197C2211yescharacteristic, complemented
31#95×C197C2212yescomplemented
32C24×C197—28no—
33C24×C197—24no—
34C24×C197—24no—
35C24×C197—24no—
36C24×C197—44no—
37C24×C197C2314yes—
38C24×C197C2311yescharacteristic
39C24×C197C2311yescharacteristic
40C2×D4×C197—232no—
41C2×D4×C197—22no—
42C2×D4×C197—28no—
43C2×D4×C197—22no—
44C2×D4×C197—28no—
45C2×D4×C197—28no—
46C2×D4×C197—24no—
47C2×D4×C197—24no—
48C2×D4×C197—24no—
49C2×D4×C197—44no—
50C2×D4×C197D418yescomplemented
51C2×D4×C197D414yescomplemented
52C22×C4×C197—24no—
53C22×C4×C197—22no—
54C22×C4×C197—24no—
55C22×C4×C197D414yescomplemented
56C22×C4×C197C2314yes—
57C22×C4×C197C2311yescharacteristic
58C22×C4×C197C2311yescharacteristic
59C22×C4×C197C2311yescharacteristic
60C22×C4×C197C2311yescharacteristic
61C22×C4×C197C2311yescharacteristic
62C4⋊C4×C197—22no—
63C4⋊C4×C197—24no—
64C22⋊C4×C197—22no—
65C22⋊C4×C197—24no—
66C22⋊C4×C197—28no—
67C22⋊C4×C197—22no—
68C22⋊C4×C197—28no—
69C22⋊C4×C197—44no—
70C42×C197—22no—
71C23×C197—22no—
72C23×C197—22no—
73C23×C197—216no—
74C23×C197—21no—
75C23×C197—24no—
76C23×C197—22no—
77C23×C197—24no—
78C23×C197—24no—
79C23×C197—21no—
80C23×C197—24no—
81C23×C197—24no—
82C23×C197—22no—
83C23×C197—24no—
84C23×C197—416no—
85C23×C197—416no—
86C23×C197—48no—
87C23×C197—48no—
88C23×C197C2×D418yes—
89C23×C197C2×D412yes—
90C23×C197C2×D412yes—
91C23×C197C2×D412yescomplemented
92C23×C197#4112yes—
93C23×C197C2411yescharacteristic
94D4×C197—216no—
95D4×C197—28no—
96D4×C197—216no—
97D4×C197—416no—
98C2×C4×C197—21no—
99C2×C4×C197—22no—
100C2×C4×C197—21no—
101C2×C4×C197—28no—
102C2×C4×C197—21no—
103C2×C4×C197—22no—
104C2×C4×C197—24no—
105C2×C4×C197—21no—
106C2×C4×C197—24no—
107C2×C4×C197—22no—
108C2×C4×C197—24no—
109C2×C4×C197—44no—
110C2×C4×C197C2×D412yes—
111C2×C4×C197C2×D412yes—
112C2×C4×C197C2×D414yes—
113C2×C4×C197C2×D412yescomplemented
114C22×C197—22no—
115C22×C197—22no—
116C22×C197—24no—
117C22×C197—24no—
118C22×C197—28no—
119C22×C197—22no—
120C22×C197—24no—
121C22×C197—28no—
122C22×C197—28no—
123C22×C197—432no—
124C22×C197—41no—
125C22×C197—44no—
126C22×C197—41no—
127C22×C197—44no—
128C22×C197—44no—
129C22×C197—42no—
130C22×C197—44no—
131C22×C197#12114yescomplemented
132C22×C197C22×D411yescharacteristic
133C22×C197C22×D412yes—
134C22×C197C22×D411yescharacteristic
135C22×C197C2×#4111yescharacteristic
136C22×C197H(5,𝔽2)12yes—
137C4×C197—22no—
138C4×C197—22no—
139C4×C197—41no—
140C4×C197—41no—
141C4×C197—44no—
142C4×C197#12014yescomplemented
143C2×C197—24no—
144C2×C197—42no—
145C2×C197—48no—
146C2×C197C2×#12011yescharacteristic
147C2×C197C2×#12111yescharacteristic
148C2×C197#53511yescharacteristic
149C2×C197D4212yes—
150C2×C197#54612yes—
151C197#240811yescharacteristic, direct product factor, Sylow
152#2408C19711yesmaximal, characteristic, direct product factor, Sylow
153C23×D4C2×C19711yescharacteristic, complemented
154C2×#121C2×C19712yescomplemented
155C2×#120C2×C19712yescomplemented
156C2×C4×D4C2×C19711yescharacteristic, complemented
157#394C2×C19712yescomplemented
158#392C2×C19712yescomplemented
159#390C2×C19712yescomplemented
160#380C2×C19711yescharacteristic, complemented
161#379C2×C19712yescomplemented
162C25C22×C19712yes—
163C22×D4—28no—
164C22×D4—24no—
165C22×D4C22×C19712yes—
166C22×D4C22×C19711yescharacteristic
167C22×D4C22×C19712yes—
168C22×D4C22×C19714yescomplemented
169C23×C4C22×C19711yescharacteristic
170C23×C4C22×C19711yescharacteristic, complemented
171#121—28no—
172#120—28no—
173C4×D4—24no—
174C2×C4⋊C4C22×C19711yescharacteristic, complemented
175C2×C4⋊C4C22×C19712yescomplemented
176C2×C22⋊C4C22×C19711yescharacteristic
177C2×C22⋊C4C22×C19712yes—
178C2×C22⋊C4C22×C19711yescharacteristic, complemented
179C2×C22⋊C4C22×C19714yescomplemented
180C2×C22⋊C4C22×C19714yescomplemented
181C2×C22⋊C4C22×C19714yescomplemented
182C2×C42C22×C19711yescharacteristic, complemented
183#95C22×C19712yescomplemented
184C24—28no—
185C24—24no—
186C24—24no—
187C24—24no—
188C24—44no—
189C24C23×C19714yes—
190C24C23×C19711yescharacteristic
191C24C23×C19711yescharacteristic
192C2×D4—232no—
193C2×D4—22no—
194C2×D4—28no—
195C2×D4—22no—
196C2×D4—28no—
197C2×D4—28no—
198C2×D4—24no—
199C2×D4—24no—
200C2×D4—24no—
201C2×D4—44no—
202C2×D4D4×C19718yescomplemented
203C2×D4D4×C19714yescomplemented
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—24no—
207C22×C4D4×C19714yescomplemented
208C22×C4C23×C19714yes—
209C22×C4C23×C19711yescharacteristic
210C22×C4C23×C19711yescharacteristic
211C22×C4C23×C19711yescharacteristic
212C22×C4C23×C19711yescharacteristic
213C22×C4C23×C19711yescharacteristic
214C4⋊C4—22no—
215C4⋊C4—24no—
216C22⋊C4—22no—
217C22⋊C4—24no—
218C22⋊C4—28no—
219C22⋊C4—22no—
220C22⋊C4—28no—
221C22⋊C4—44no—
222C42—22no—
223C23—22no—
224C23—22no—
225C23—216no—
226C23—21no—
227C23—24no—
228C23—22no—
229C23—24no—
230C23—24no—
231C23—21no—
232C23—24no—
233C23—24no—
234C23—22no—
235C23—24no—
236C23—416no—
237C23—416no—
238C23—48no—
239C23—48no—
240C23C2×D4×C19718yes—
241C23C2×D4×C19712yes—
242C23C2×D4×C19712yes—
243C23C2×D4×C19712yescomplemented
244C23#41×C19712yes—
245C23C24×C19711yescharacteristic
246D4—216no—
247D4—28no—
248D4—216no—
249D4—416no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—21no—
253C2×C4—28no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—24no—
257C2×C4—21no—
258C2×C4—24no—
259C2×C4—22no—
260C2×C4—24no—
261C2×C4—44no—
262C2×C4C2×D4×C19712yes—
263C2×C4C2×D4×C19712yes—
264C2×C4C2×D4×C19714yes—
265C2×C4C2×D4×C19712yescomplemented
266C22—22no—
267C22—22no—
268C22—24no—
269C22—24no—
270C22—28no—
271C22—22no—
272C22—24no—
273C22—28no—
274C22—28no—
275C22—432no—
276C22—41no—
277C22—44no—
278C22—44no—
279C22—41no—
280C22—44no—
281C22—44no—
282C22—42no—
283C22#121×C19714yescomplemented
284C22C22×D4×C19711yescharacteristic
285C22C22×D4×C19712yes—
286C22C22×D4×C19711yescharacteristic
287C22C2×#41×C19711yescharacteristic
288C22H(5,𝔽2)×C19712yes—
289C4—22no—
290C4—22no—
291C4—41no—
292C4—41no—
293C4—44no—
294C4#120×C19714yescomplemented
295C2—24no—
296C2—48no—
297C2—42no—
298C2C2×#120×C19711yescharacteristic
299C2C2×#121×C19711yescharacteristic
300C2#535×C19711yescharacteristic
301C2D42×C19712yes—
302C2#546×C19712yes—
subgroupG/H
center, socleC23×C197C24
commutator, Frattini, cosocleC23C24×C197
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C197 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C23×D4×C197C2semidirect
2. C2×#121×C197C2semidirect
3. C2×#120×C197C2semidirect
4. C2×C4×D4×C197C2semidirect
5. #394×C197C2semidirect
6. #392×C197C2semidirect
7. #390×C197C2semidirect
8. #380×C197C2semidirect
9. #379×C197C2semidirect
16. C22×D4×C197C22semidirect
18. C23×C4×C197C22semidirect
22. C2×C4⋊C4×C197C22semidirect
23. C2×C4⋊C4×C197C22semidirect
26. C2×C22⋊C4×C197C22semidirect
27. C2×C22⋊C4×C197C22semidirect
28. C2×C22⋊C4×C197C22semidirect
29. C2×C22⋊C4×C197C22semidirect
30. C2×C42×C197C22semidirect
31. #95×C197C22semidirect
50. C2×D4×C197D4semidirect
51. C2×D4×C197D4semidirect
55. C22×C4×C197D4semidirect
91. C23×C197C2×D4semidirect
113. C2×C4×C197C2×D4semidirect
131. C22×C197#121semidirect
142. C4×C197#120semidirect
151. C197#2408direct (N × H)
152. #2408C197direct (N × H)
153. C23×D4C2×C197semidirect
154. C2×#121C2×C197semidirect
155. C2×#120C2×C197semidirect
156. C2×C4×D4C2×C197semidirect
157. #394C2×C197semidirect
158. #392C2×C197semidirect
159. #390C2×C197semidirect
160. #380C2×C197semidirect
161. #379C2×C197semidirect
168. C22×D4C22×C197semidirect
170. C23×C4C22×C197semidirect
174. C2×C4⋊C4C22×C197semidirect
175. C2×C4⋊C4C22×C197semidirect
178. C2×C22⋊C4C22×C197semidirect
179. C2×C22⋊C4C22×C197semidirect
180. C2×C22⋊C4C22×C197semidirect
181. C2×C22⋊C4C22×C197semidirect
182. C2×C42C22×C197semidirect
183. #95C22×C197semidirect
202. C2×D4D4×C197semidirect
203. C2×D4D4×C197semidirect
207. C22×C4D4×C197semidirect
243. C23C2×D4×C197semidirect
265. C2×C4C2×D4×C197semidirect
283. C22#121×C197semidirect
294. C4#120×C197semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C23×D4×C1972
C2×#121×C1972
C2×#120×C1972
C2×C4×D4×C1972
#394×C1972
#392×C1972
#390×C1972
#380×C1972
#379×C1972
#2408197